Time logging helps you keep track of the time you spend on a specific task.
When a task gets assigned to you, the "Log time" button enables and you could click on it to activate the time tracker . At this point you can close the modal, or even close the browser for that matter. We'll keep track of the time for you so you can get to work.
You can see the amount of tracked time since you last clicked on "Log time" on the task in "Current time", and the total amount of time tracked by all task members in "Total time" section of the task sidebar.
Once you've finished working on that task, just click on the logger to finish time tracking.
Another way of accessing your timelogs is through the Timelogs section on the sidebar on the right side of the page. You could open each task to stop the timelog, or you could just press stop button to stop them all.